A square matrix is a unit in the matrix ring iff its determinant is a unit in the ring.
square matrices are endomorphisms: self-maps from X→X i.e. X⟲. Sheldon Axler tells us that this is the most important class of linear map
linear is a crucial gap-of-understanding between mathematicians and non-. I used to think "nonlinear" was cool because, ya know, chaos and turbulence and magical wizard-brains. Homomorphisms let you probe one space with another thing -- like probing a surface with closed loops. "Linear" in this sense just means preserving the relevant facts about the first space, when it’s sent into / onto the second space. Hysterical invocations of "nonlinearity" or "nonsmoothness" sometimes indicate that you needed to look at a closely related space where things are linear (simple pieces --- or for example this is the whole reason people like the flat tangent-bundle world of derivatives/pushforwards and tangent spaces in Lie theory) , sometimes that you should have broken the nonlinear map into a short sequence of stupider maps, and sometimes that you have no structure at all and thus can make no
a ring happens when you put two operations together. (Each of the operations should be a group, i.e. associative, invertible, with identity. And give yourself the .) Rings don’t necessarily have «division» in the usual sense, so you can’t always solve ab=ac down to b=c. For example in clock arithmetic with a 12-hour cycle, 3•4 = 6•4 = 12 = 0, but 3 o'clock ≠ 6 o'clock. The groups could be familiar integer operations like + or × on ℤ, they could be geometrical like reflections of a geometric figure](quora), or they could be permutation groups of the three letters ABC ↦ {CAB, BCA, etc.}
square matrices are boxes filled up with numbers, and given this multiplication rule: . The numbers can come from different worlds; if they come from a world where ÷ makes sense, that could be different from a "ring"-world where ÷ might not make (the usual kind of) sense.
